As long time readers of this blog know, one of the things that drive me crazy – in fact, one of the things that led me to start this blog – is the rampant innumeracy of our society. The vast majority of
Americans have no real knowledge or comprehension of numbers or mathematics, and what makes that even worse is that most really, truly, fundamentally don’t care.

A vivid example of that is demonstrated in a recent Supreme Court ruling in a case dealing with the use of sonar in submarine training
by the US navy in waters inhabited by whales.

The basic idea behind the case is that environmental groups had sued to prevent the navy from doing Sonar tests and training in waters where they were likely to harm marine mammals.

Before getting to the innumeracy, it’s interesting to just
take a look at the basic idea of the case, and how some basic numbers fit into it.

Sonar is not an innocuous technology. Many of us have intuitions
about it taken from movies and television shows, where it’s just a “ping” sound. But the point of sonar is to create a sound wave in the water that is powerful enough to produce accurately measurable reflections off of bodies at great distances away. Because of the way that sound propagates, there’s an inverse square relationship between signal strength and distance. So if you’ve got a target 100 meters
away, you’ve first got an inverse square reduction in the strength of the signal before it even hits the target. Then part of the
signal is absorbed by the target, and part is reflected. The reflected part again diminishes by inverse square. The point here is that
it takes a very powerful sound pulse to get good readings.

A typical sonar pulse originates at a volume of around 235 decibels.
That is loud. That is amazingly loud. In fact, that’s louder
that the loudest possible sound in air by roughly 16 times! (I
originally wrote 4 times, when it fact it’s four doublingsin volume.)

To put that into context, it’s worth looking at a couple of examples to give you a sense of how much energy we’re talking about in a sonar pulse.

If you’re standing on a platform on the NYC subway, a
train arriving in a station is around 100 decibels on the platform.

If you’re standing 100 feet away from the jet engine on a Boeing
737 when it’s pushing the plane up the taxiway, the sound of the
engine will be roughly 140 decibels.

Standing immediately in front of the speaker towers at a rock concert is about 150 decibels.

180 decibels will instantaneously rupture your eardrums.

In air, the maximum possible loudness of sound carried through the atmosphere at typical air pressure is about 195 decibels.

Decibels are logarithmic – each 10 decibel increase corresponds roughly to a doubling of volume, and volume pretty much corresponds to the amount of energy packed into the sound wave.

So when we’re talking about sonar, we’re talking about a bloody hell of a lot of energy being pumped into the water. It’s a concussive wave of immense force. There are well documented instances of sonar pings causing bleeding around the eyes and ears of whales; there’s also some poorly understood data showing high correlation between high-energy sonar use and whale beachings.

The Navy policy is to not do Sonar tests if marine mammals were sighted within 200 yards. The Natural Resources Defense Fund sued the Navy to try to force them to not do Sonar tests if marine mammals were sighted within 2000 yards.

It’s a damn big difference. Given the inverse-square relationship between the power from the sonar ping at a point, and the distance away from the source of that point, you’re talking about reducing the maximum
exposure to Sonar waves by more than 30 decibels – a very significant
difference!

Now, on to the innumeracy: In his decision, Chief Justice Roberts
wrote:

The District Court’s injunction does not include a graduated power-down,
instead requiring a total shutdown of MFA sonar if a marine mammal is
detected within 2,200 yards of a sonar-emitting vessel. There is an
exponential relationship between radius length and surface area (Area =
πr2). Increasing the radius of the shutdown zone from 200 to 2,200 yards would accordingly expand the surface area of the shutdown zone by a factor of over 100 (from 125,664 square yards to 15,205,308 square yards).”

Where’s the problem? The surface area affected by a given sighting range is, as he points out, a circle with area πr2. But that’s a polynomial expression, not an exponential one. We’re not talking about a trivial distinction here.

So the problem is the word “exponential”. It doesn’t mean what he thinks it means. You might say that I’m just being pedantic here – so what if he got a word wrong?

But he’s a judge. His career is in studying laws, legal
judgements, and carefully piecing apart the precise meanings of those
words. If you’ve ever known an appeals judge, you’ll know that they’re
incredibly pedantic about precise meanings of terms – they have to be to
do the job. So we’re talking about a guy who’s supposedly an expert on
language semantics, at least where legal issues are involved. If anyone were to make a legal filing where they used the word “libel” where they meant “liable”, he’d throw the case out on it’s ass.

And he clearly thinks that he’s showing off, by including
an equation and fancy mathematical words in his judgement. But he can’t be bothered to actually understand the meanings
of the words that he’s using.

And those words mean something very different. Again, it helps to look at a simple example to compare. If we use 2 as our base, in the polynomial expression πr2, the
change from a 200 yard radius to a 2200 yard radius is different
by a factor of 121 times. If we used an exponential expression based on 2, π2r, we’d get a difference factor of, roughly,
1.15×10604.

That’s the difference between a number that my five year-old son can count to in about three minutes, and a number that’s absolutely unimaginable – something so large that saying that it’s more than millions of times larger than the number of particles in the entire universe doesn’t begin to approach it.

Hell, replace the “2” with something smaller in the exponential;
say, 1.01. That’s something that’s going to increase very slowly
according to an exponential curve, right? It’s barely more than 1.
Raising it to the fifth power only gives you 1.05 – it’s a very
shallow exponential curve. But take the 200 versus 2200 from the
lawsuit, and you’ll find that the ratio of the exponential, π1.01r compared to the quadratic polynomial πr2 is, roughly, 439 million.

Big difference. And the fact that that difference, between an equation with an exponent, and a exponential equation is totally lost on the chief justice of the U. S. Supreme Court is just pathetic.

(On the other hand, the person who originally sent this to me – and several folks in comments in the newspapers that carried this story – claimed that πr2 is the wrong equation, because you should be considering the surface of a sphere. Alas, the relevant regulations are for sighting marine mammals within a given radius on the surface That’s a bit of a silly confusion too – a nuclear submarine typically never goes below about 1600 feet; their hulls will collapse from pressure at around 2400 feet; there’s just no way that a spherical space makes sense. Submarines go beneath the water for stealth reasons, but they can’t go very deep at all; one half mile, and they’ll be crushed like a tin can. No one would really talk about a spherical area around a sub; the way they operate, it’s got no value.)

Jason: I think we can give Chief Justice Roberts a bit of slack on that term, since he was concerned about the area on the surface of the water that would now be excluded. So “surface area” means a section of the earth’s surface area, and for that \pi r^2 is a close enough approximation given the distances involved.

Marc: Unfortunately, the kind of innumeracy judges are subject to is the kind that gets coded into dictionary definitions. Justice Robert’s mangling would almost certainly be consistent with definition (1) given at that link.

It used to drive me nuts when I was employed tutoring calculus students and one of them insisted on using the verb “derive” when he meant “take the derivative of”. As in: “So then I derive x-squared and get two x”.

Slight correction: you say that the maximum sound pressure in air is 195 dB, and sonar pings are generated at 235 dB. That should be 10000 times as powerful, I believe, not a mere 4 times, since decibels are base-10 logarithmic.

(Since you always say “loud”, I’m not sure if you’re silently accounting for the fact that increases in sound pressure are translated to increases in subjective loudness non-linearly. If so, it might be a good idea to mention that.)

OK, so let’s apply your logic with micah’s comment: on this evidence, you apparently don’t know the difference between “N times greater than” and “N order of magnitudes greater than”. (Of course, we know you do really and this was just a slip!)

A judge refers to a correct formula with an exponent in as exponential, you say “nyaa nyaa it’s polynomial”. Who is more wrong? The judge with the correct formula and a mildly misused term or the professional mathematician who apparently doesn’t understant some basic concepts?

If the energy originates at a point and spreads out equally in all directions in a spherical wavefront, then the energy absorbed by a given area (ear drum) at a given distance (r) would vary by 1/(r^2), correct? Logically this is because the energy isn’t distributed evenly through the volume it’s traversed, but is rather ‘focused’ at the leading edge (the shell of the sphere).
However, that assumes that the energy spreads equally in all directions and none is reflected (by the air/water interface, or the bottom or other structures). Also I was assume that for many applications of sonar, focusing/directionalizing the sound would be done, if possible to allow a larger range with a lower total power input. Were these considerations taken into account?

Are you sure? All the internet sources I can find use base 10 logarithms (or, to be more precise, base 101/10 logarithms) in their definition of decibels. I don’t have a physics textbook handy to confirm, but…

Mark, there are some additional errors here, which have less to do with innumeracy than with ignorance of acoustics.

SPL expressed in decibels is not directly comparable between the air and water media.

For sound propagating in air, the reference 0 dB SPL pressure is 20 uPa RMS, which corresponds to an intensity (power density) of 10-12W/m2. For sound in water, the O dB reference pressure is defined as 1 uPa, which corresponds to an intensity of approximately 6 x 10-19W/m2, due to the fact that the characteristic acoustic impedance of water is approximately 3400 times that of air.

So, one has first to deal with the different 0 dB reference levels, and also to ask which is the truly relevant quantity- pressure or power density.

It might also be worth mentioning that the inverse square law applies only to a spherically diverging wave. If the divergence is constrained to 0 in both the horizontal and vertical dimensions, as in a plane-wave tube, the SPL remains unchanged with distance save for losses due to viscous effects in the medium. If it’s constrained in only one dimension, e.g., by the surface and bottom of the ocean, the divergence becomes cylindrical at a distance comparable to the depth of the water, in which case the power/distance relationship is inverse linear.

First, as I suspect you know, decibel are meaning just a ratio, so it’s meaning less unless the reference level is either specified or infer able from context.

Sound pressure level in water and air use different reference pressures, in air 20uPa is used while in water 1 uPa is commonly used, giving a dB SPL number 26 dB greater in water relative to air. I’ve also seen mPa used, a 34 dB difference, which means for the same pressure add 34 dB to the SPL level in air.

Second, even accounting for this the threshold of human hearing in water is 67 dB (ref 1uPa, see wikipedia “Sound Pressure”).

And one last this to consider is the overpressure, deltaP/P, a wave at 200 dB SPL ref 1uPa is 10,000 Pa. Taking 100 m as the depth we are interested in the pressure is 980,665 Pa giving an overpressure of just over 1%. And is the equivalent of a 1m wave passing over you on the surface.

I’m still not sure what the sum total effect of all these additional complications are. I’ll still agree that 235 db SPL re ??? in water from most any reference is crazy loud and not anywhere I want to be even remotely close to.

Ktesibos strikes a key point here in the difference between 0 dBA and 0 dB re 1uPa. In those units it is then also relevant that casual whale conversation uses signal levels that are ~180 dB re 1uPa @ 1 m from the source (these are at similar mid-frequencies) and uses pulses to probe the near environment (higher frequency) that can be as loud as 220 dB. Does MF sonar cause whales harm? Perhaps but it is not analogous to being acoustically blasted by a jet engine. If anything it is analogous to being scared by distant air raid siren and darting into traffic.

Why do lawyers (and judges) and diplomats and doctors and pharmacists use Latin terminology?

Because Latin is a dead language. Hence these words do NOT change their basic meaning over time, the way that words in English, French, German, Mandarin, Arabic, Spanish, Portuguese, and the like do.

Appeals (State or Federal Distict) are much more expensive and procedurally exacting than Superior Court cases. Supreme Court (State or US) are much more expensive and procedurally exacting than Appellate cases.

Theoretically, this is due to the hierarchy associated
with Stare Decisis.

Economically, it is the iron law of Supply and Demand.

There is a smaller supply of lawyers who have experience in Appeals. Still smaller is the fraction who’ve argued in Supreme Court.

The demand on their time is higher.

Thus, the price that they charge is higher.

Now, more theory.

[Disclaimer: IANAL: I Am Not A Lawyer]
[Disclaimer: TINLA: This Is Not Legal Advice]

What we call “The Law” (State and Federal courts) is a chaotic attractor in the space of all possible laws. Law, being precedent-based rather than axiom-based (as in Math), or Empirical basis (Physics experiments) have a sensitivity to initial conditions (i.e. precise language of legislation) and feedback mediated by Stare decisis which [United States Internal Revenue Serv. v. Osborne (In re Osborne), 76 F.3d 306, 96-1 U.S. Tax Cas. (CCH) paragr. 50,185 (9th Cir. 1996)] is the policy of the court to stand by precedent; the term is but an abbreviation of stare decisis et quieta non movere = “to stand by and adhere to decisions and not disturb what is settled.” Consider the word “decisis.” The word means, literally and legally, the decision. Nor is the doctrine stare dictis; it is not “to stand by or keep to what was said.” Nor is the doctrine stare rationibus decidendi = “to keep to the rationes decidendi of past cases.” Rather, under the doctrine of stare decisis a case is important only for what it decides — for the “what,” not for the “why,” and not for the “how.” Insofar as precedent is concerned, stare decisis is important only for the decision, for the detailed legal consequence following a detailed set of facts.

As wikipedia begins:

John Glover Roberts, Jr. (born January 27, 1955) is the seventeenth and current Chief Justice of the United States. Appointed by President George W. Bush, Roberts is generally considered a member of the more conservative wing of the court.

Before joining the Supreme Court on September 29, 2005, Roberts was a judge on the United States Court of Appeals for the District of Columbia Circuit for two years.

Previously, he spent 14 years in private law practice and held positions in Republican administrations in the U.S. Department of Justice and Office of the White House Counsel.

Thank you very much President George W. “Most Anti-Science President Ever” Bush. Your legacy of ignorance and superstition continues unabated.

Good thing that President Obama will replace some Supreme Court Justices. The sooner the better.

Litmus Test: must believe that this is the real world, which is bound by Physical Laws that trump man-made laws, and which follow Mathematical equations.

Nobody caught this bit – as an audio engineer at times, I had to know this: 3db is a doubling of the signal, not 10db. 10 db is doubling three times (approx) therefore a factor of 8+, not 2. The nonlinear compressive response of the ear makes for an un-reliable measuring device.

1) The \pi r^2 formula IS the result from considering the surface of a sphere. Release x units of energy at time t and at time t’ it will be (approximately) distributed over some sphere of some radius. Thus the effect at distance r is proportional to 1/r^2 because that’s the ratio between the radio and SURFACE AREA.

However, it does make sense to use a sphere since the fact that the subs don’t dive very deep has nothing to do with whether energy is carried off into the depths of the ocean.

——–

2) We (mathematicians) are largely complicit in this innumeracy, ironically, through our ham handed attempts to demand students at least know about exponential equations or what not. The reason so many students are so unbelievably bad at math is that the years and years or rote work without useful application or understanding they hate the subject and have learned not to try and understand it.

Frankly, we should change the name of the subject. Let math be (highschool) algebra and call the university subject “mathematical sciences” or something. Then on top of that absolutely refuse to teach courses that demand computation without understanding.

—-

Also, while I don’t agree there is this unpleasant trend toward using the word exponential to mean big, including by many people who obviously DO know what it really means.

After all, who needs the Axiomatic Truth of Mathematics, or the Empirical Truth of Science, when they have a direct access to Revealed Religious Truth, from a “higher Father” who tells our President to invade Iraq? The entire world knows that the USA is a wacko Theocracy, like Iran…

UNITED NATIONS, Nov. 13 — Employing unusually vivid religious imagery for the secular United Nations, President Bush on Thursday praised the “transformative and uplifting power of faith” and said religious belief “leads us to common values.”

Addressing a two-day interfaith conference that has prompted mixed reactions from other leaders, Bush said religious belief “changed my life” and “sustained me through the challenges and joys of my presidency.” He also suggested faith can transform relations between nations and cultures.

“One of my core beliefs is that there is an almighty God, and that every man, woman and child on the face of this Earth bears his image,” Bush said. “. . . I know many of the leaders gathered in this assembly have been influenced by faith as well. We may profess different creeds and worship in different places, but our faith leads us to common values.”
[truncated]

It used to drive me nuts when I was employed tutoring calculus students and one of them insisted on using the verb “derive” when he meant “take the derivative of”. As in: “So then I derive x-squared and get two x”.

Erm… in German, that’s standard terminology as used by math teachers and textbooks. As in “I derive x² + y after x”.

If your entire rant is based on someone in the public space using the word “exponential” when the math is polynomial, then yes, you are being a pedant. Exponential is used very often for all sorts of growth, and the fact it has a very precise meaning for mathematicians does not mean the word has the definition in popular usage. Similarly, mathematicians use phrases like “blowing up” and “undecidable” which have different meanings to mathematicians versus lay-people. When lay-people say something is “blown up” or “undecidable” you can be sure they are not talking about resolving singularities on algebraic varieties or the intricacies of Godel’s arguments. Similarly “exponential” in common usage does *not* have to match your precise mathematical usage.

As someone posted above, dictionaries are consistent with what SCOTUS wrote. Your limited (yet precise) definition of exponential is the problem.

“Exponential growth” means that it always takes the same amount of time (or distance) for a quantity to multiply itself by the same amount. For instance, if something doubles in size every 20 minutes, that is exponential growth. (It will also take always the same amount of time to treble in size, or to increase tenfold.)

Just because people misuse a word, should not dilute its precise meaning.

The cited case of innumeracy had nothing to do with the Supreme Court’s decision anyway — the truth of the matter was that a District Court had overstepped its authority by trying to restrict the Navy’s use of sonar. This was a national security issue that wound up having little to do with math, or with environmental impact for that matter.

So what you’re saying is that the innumeracy inserted into the court’s opinion was in fact entirely gratuitous innumeracy? I think that makes it worse; people being flagrantly innumerate when they need to be is a sad thing, but flagrant innumeracy that’s also completely unnecessary?

I work for an state environmental agency, and spend some of my time running the noise pollution program. If you are frustrated by innumeracy related to acoustics in this case – just imagine trying to explain decibels and noise rules to the rural dairy farmer or retiree with too much time on their hands.

Glad to see the case, acoustics, and innumeracy discussed here – lots of good points in the comments, too!

What a bafoon! He said “exponential” when he should have said “polynomial”. Can you believe what a retard the Supreme Court Justice is? He doesn’t even know that in an exponential equation the exponent is a variable. What an idiot. Just because an equation has an exponent doesn’t make it an exponential equation. Can you believe this innumeracy? This is really important!!! How can you take a Supreme Court decision seriously if such a simple set of terms is misused in this manner? This is going to be in the public record forever. Do you hear me? Forever! This reflects poorly on the entire United States, our education system, and our judiciary. Exponential. Polynomial. Learn the difference people. Don’t be like Justice Roberts. I bet Ginsburg wouldn’t have made that kind of error.

OT but a particularly irritating form of innumeracy is use of “crescendo” to mean dloudness/dt=0, d2loundness/dt2 less than 0. Don’t do it people. A crescendo is dloudness/dt greater than 0. You will know a crescendo when you reach one because it starts softly. /rant

Nope, it’s a concussive wave of immense pressure. This is not a trivial distinction. Physics illiteracy, a mathematician should know better, etc ad nauseum.

See how silly this is? A one-word error in math terminology is not a huge issue. It doesn’t affect the reasoning at all, as in fact the rest of the paragraph actually involving the numbers is just fine. A person might as well leave the room in a huff every time a professor accidentally drops a sign in doing a math problem.

Daithi raises interesting point. Read somewhere that bats have got some fancy mechanism in their ears to stop them deafening themselves when they scream. Wonder if whales have anything similar. Any biologists out there?

Samuel A. Alito, Jr., once confirmed, became the fifth Catholic on Supreme Court simultaneously. On 31 October 2005, as a Halloween trick on the nation, Pfter President George W. Bush announced his nomination of Samuel Alito to replace outgoing justice Sandra Day O’Connor on the U.S. Supreme Court. At the time of his nomination, Alito was a judge on the United States Court of Appeals for the Third Circuit whose ideological similiarities to U.S. Supreme Court Justice Antonin Scalia had earned him the nickname “Scalito.”

The four Catholics whom he joined were Chief Justice John Roberts and Associate Justices Antonin Scalia, Clarence Thomas, and Anthony Kennedy, the latter of whom has been a disappointment to Catholics who hoped he would help overturn Roe v. Wade or uphold legal restrictions on abortion.

Alito is the 11th Catholic in U.S. history to sit on the Supreme Court and the current court has for the first time a majority of Catholics on the nine-member court.

Given the statistical distribution of relgious believers in the USA, whar is the a prior probability of 5/9 of the court being of the same religion?

I am extremely uncomfortable with a majority being co-religionists. I would be just as uncomfortable if the majority were Protestant, or Jewish, or Buddhist, or Hindu, or Mormon, or Islamic, or whatever. I’m not even sure I’d be happy if a majority were atheists.

Doesn’t this raise problems of bias, and can it be detected in the correlation coefficients? There have already been publications that show an orthonormal basis of the decisions to be far less than 9 dimensions. I vaguely recall it as being around 4.

May I respectfully suggest you’ve missed the forest for the semantic trees? The passage you’ve cited from the SCOTUS opinion is very simply a cheap attempt by the Chief Justice to convert “Shut the sonar down if a whale is spotted at a distance 10 (11?) times further away than what you’ve been using,” to “OMFG! Do you know what a huge area that is?!!”

Yeah, and if you converted it to a volume difference between spheres, it would look bigger yet. Big deal. The pedantic stuff about misuse of mathematical terms completely misses the really significant point that the Chief Justice is just multiplying more numbers together, for no particular scientific reason, to make the lower court’s order look unreasonable.

Given the statistical distribution of relgious believers in the USA, whar is the a prior probability of 5/9 of the court being of the same religion?

Quite high, I think; although I don’t have exact numbers at my fingertips, I think somewhere in the neighborhood of 70-80% of the USA is the same religion (Christian). In fact some religious minorities are de facto excluded on the basis of their beliefs (atheists, and at the present moment, probably Muslims, and possibly other non-Christian non-Jews as well), which drives the probability even higher.

I don’t know if there’s been any time in the Court’s history that it hasn’t been majority Christian. It’s only if you insist on counting Christianity as more than one religion that the phenomenon can look unusual at all.

Also, I agree with Jud: Roberts is using numbers for rhetorical effect, not to inform anyone of anything.

I’m going to have to agree strongly with #39. It’s really there only as an impressive looking (but really rather weak) rhetorical device.

In fact, I would argue that the statement that the sonar needs to be shut off if a whale is within a 15,205,308 square yard patch instead of a 125,664 square yard patch contains less useful information than saying the sonar needs to be shut down if the whale is closer than 2200 yards instead of 200 yards, since range is really the more relevant quantity in this case. IMO the whole statement is out of place and irrelevant.

The intensity-range relation in underwater acoustics
depends on many things. Exponents of -2, -1.5, and -1
(and others in between) can be good fits depending on
frequency and the local environment. The idea of a single
context-independent sighting limit is far more basically
innumerate than the judge’s mistake.

I must point out that you’ve made a minor error of terminology in your post! You talk about “the number of particles in the entire universe”… but don’t you mean the number of atoms in the observable universe?

I’ve never heard an argument that dark matter and dark energy particles (if such things exist) shouldn’t outnumber baryons by hundreds of orders of magnitude*, so you should say “atoms” (or “protons,” since most atoms are of hydrogen) instead of “particles.”

More importantly, the number of atoms in the entire universe could be infinite, for all we know. The observable universe is a finite and perfectly well-defined part of the universe—the only part of the universe about which we currently have any right to speak!

* Although there might be one, based on the assumption that we can rule out the existence of particles below a certain mass just because they haven’t been produced (as far as we can tell!) in particle accelerators.

One of my low moments of despair in this regard occurred one morning on the way to work when a newscaster for our local NPR station blithely reported that a town of “ten thousand” no longer existed on the plains of Kansas because a brush fire had burned it all down.

A town of ten thousand on the Kansas plains? Burned to the ground? And it was just an add-on at the end of the news cycle?

Whales are so beautiful singers and so intelligent and social beings. It makes me very sad to imagine how they get terrified and harmed by this. There was a physics lession by Prof. Muller from MIT, I watched it on Google Video, where it was explained that there are special water depths in the ocean where sound can travel with almost no damping, so very long distance. It is because of some reflection/lensing effect due to the depth-increasing water-pressure. Whales and Dolphins might even use this for global communication in my opinion. It was also used in World War II for finding crashed fighter-pilots, they had some small but heavy, hollow metal balls with them which would sink, implode at a certain water level and send a detectable sound signal to some stations near the coast. Anyway. There is story in Aboriginal Australia… you know every tribe there has some special animal which is taboo to them from the Dreamtime on, because it is their brother and they perform dedicated songs… the story goes that there is one whale swimming in the ocean carrying the human song, the very special song that defines and preserves what we are as human beings. Feel this in your heart! Another kind of truth than Mathematics maybe… sorry if “Bad Math”. Marcus

As many commentors pointed out already this particular blog botches English, Math, and Physics while trying to point out how a conservative Supreme Court justice is wrong, trivially but fundamentally wrong. Red flags abound.

Oh well, the point Mark tried to make is that “polynomial” is a better fit than “exponential”. But it’s not. The term “polynomial” would have been ambiguous as it describes both relatiopships being contrasted. It would not make sense to say that the relationship is different because it is ‘polynomial’.
Then the justice was perfectly clear having even given the formula.

Most importantly though is that this topic has nothing to do the overall decision. Mark is claiming a small error in something inconsequential. That is pedantic.

Neil B: This, to me, is different from the response that Marilyn vos Savant (not a relative of mine, despite the similar middle name) got from ‘professional statisticians’ when she correctly proposed the answer, ‘switch’, to the [Monte Hall] three-door problem.”

That’s because Mathematicians may differ, but the Mathematical community eventually settles on a consensus interpretation.

On the other hand, the American legal system is innately hierarchical, hence my comment in #16 on Stare Decisis.

The Chief Justice of the United States is at the mountain top, above all other legal authorities. He is, Constitutionally, co-equal to the President of the United States, and to the Speaker of the House. But, for legal hairsplitting, he has nobody above him.

Neither Marilyn vos Savant nor the PhDs and professors who diagreed with her were ultimate authorities.

When the ultimate authority gets things wrong, it is a whole different kettle of fish. Or of whale blubber.

Another point: the Supreme Court produces WRITTEN LAW. If wrong, this is worse than bad UNWRITTEN LAW. The ambiguities of human speech and of mathematics are likewise different, and scrambling them together is abominable.

In 1872, the advantages of {Page 13 Cal.3d 815} codification of the ***unwritten law***, as well as of a systematic revision of statute law, loomed large, since that law, drawing heavily upon the judicial traditions of the older states of the Union, was still in a formative stage. The possibility of widely dispersed popular knowledge of basic legal concepts comported well with the individualistic attitudes of the early West.” (Van Alstyne, supra, p. 6.)

… the extreme consciseness and brevity of expression which was characteristic of the 1872 code, although salutary from the point of view of popular access to basic legal concepts, early led to uncertainty and dispute as to whether it should be regarded as the exclusive or primary source of the law of private rights. Due largely to the influence of a series of articles on the subject by Professor John Norton Pomeroy, this problem of interpretation was soon resolved, and by 1920 this court was able to state with confidence: “The Civil Code was not designed to embody the whole law of private and civil relations, rights, and duties; it is incomplete and partial; and except in those instances where its language clearly and unequivocally discloses an intention to depart from, alter, or abrogate the common-law rule concerning a particular subject matter, a section of the code purporting to embody such doctrine or rule will be construed in light of common-law decisions on the same subject.” (Estate of Elizalde (1920) 182 Cal. 427, 433 [188 P. 560]; see also Van Alstyne, supra, pp. 29-35.)

In addition, the code itself provides explicit guidance as to how such construction shall proceed. “The rule of the common law, that statutes in derogation thereof are to be strictly construed, has no application to this Code. The Code establishes the law of this State respecting the subjects to which it relates, and its provisions are to be liberally construed with a view to effect its objects and to promote justice.” (Civ. Code (1872) § 4.) Also, “[t]he provisions of this Code, so far as they are substantially the same as existing statutes or the common law, must be construed as continuations thereof, and not as new enactments.”
(Civ. Code 1872) § 5; italics added.) The effect of these sections was early expressed by us in In re Jessup (1889) 81 Cal. 408, 419 [21 P. 976, 22 P. 742, 1028], in the following terms: “[E]ven as to the code, ‘liberal construction’ does not mean enlargement or restriction of a plain provision of a written law. If a provision of the code
is plain and unambiguous, it is the duty of the court to enforce it as it is written. If it is ambiguous or doubtful, or susceptible of different constructions or interpretations, then such liberality of construction is to be indulged in as, within the fair interpretation of its language, will effect its apparent object and promote justice.” (See also {Page 13 Cal.3d 816} Baxter v. Shanley-Furness Co. (1924) 193 Cal. 558, 560 [226 P. 391]; see
generally 45 Cal.Jur.2d, Statutes, § 162, pp. 663-667.)

Gilmore, Commercial Law in the United States: Its Codification and Other Misadventures, in ASPECTS OF COMPARATIVE COMMERCIAL LAW: SALES, CONSUMER CREDIT, AND SECURED TRANSACTIONS 449, 450 (J. Ziegel & W.Foster eds. 1969). In the nineteenth century, Judge Story emphasized the uncertainty inherent in legislative language:[H]uman language is, at the best, so inaccurate an instrument; there being often numerous different senses in which the same word is understood that there are, and always will be, a multitude of doubts concerning the meaning of the best drawn statutes. And all such doubts would be pure additions to those which now arise from other sources. This source of doubt does not exist in unwritten law. See Joseph Story’s Report to the Governor of Massachusetts (1836), reprinted in REPORT OF THE SPECIAL COMMITTEE TO URGE THE REJECTION OF THE PROPOSED CIVIL CODE OF THE NEW YORK BAR ASSOCIATION 36 (1882). See also J. CARTER, THE PROPOSED CODIFICATION OF OUR COMMON LAW 83 (1884): [W]henever statutory law, in its application to an unknown case, is found to work injustice in consequence of the case not having been foreseen, the effort and the tendency always is to impose violently upon the statute an interpretation not in harmony with the natural meaning of its language. In every such case, it is highly uncertain whether the effort would succeed. And this is a source of uncertainty unknown to unwritten law.

Thanks, Mark. Someone who should have known better had used the term “exponential” to describe gravity’s inverse square law. I corrected him by stating that the term should be used for a constant-to-the-power-of-variable rather than a variable-to-the-power-of-constant. His response implied that his usage was based on other people’s usage. Consistent terminology must be used to avoid confusion.

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